Static near-horizon geometries in five dimensions

We consider the classification of static near-horizon geometries of stationary extremal (not necessarily BPS) black hole solutions of five-dimensional Einstein–Maxwell theory coupled to a Chern–Simons term with coupling ξ (with ξ = 1 corresponding to supergravity). Assuming that the black holes have two rotational symmetries, we show that their near-horizon geometries are either the direct product AdS3 × S2 or a warped product of AdS2 and compact 3D space. In the AdS2 case we are able to classify all possible near-horizon geometries with no magnetic fields. There are two such solutions: the direct product AdS2 × S3 as well as a warped product of AdS2 and an inhomogeneous S3. The latter solution turns out to be a near-horizon limit of an extremal Reissner–Nordström black hole in an external electric field. In the AdS2 case with magnetic fields, we reduce the problem (in all cases) to a single nonlinear ODE. We show that if there are any purely magnetic solutions of this kind, they must have S1 × S2 horizon topology, and for ξ2 < 1/4 we find examples of solutions with both electric and magnetic fields.